Weak Galerkin Finite Element Method for the Unsteady Stokes Equation

The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H norm for the velocity, and L norm for both the velocity and the pressure by use of the Stokes projection.


Introduction
The finite element method for the unsteady Stokes equations developed over the last several decades is based on the weak formulation by constructing a pair of finite element spaces satisfying the inf-sup condition of Babuska [1] and Brezzi [2].Readers are referred to [3] [4] [5] [6] [7] for specific examples and details in the different finite element methods for the Stokes equations.The idea of weak Galerkin method was first introduced by the Professor Junping Wang in June 2011.Weak Galerkin refers to a general finite element technique for partial differential equations in which differential operators are approximated by weak forms as distributions for generalized functions.Thus, two of the key features in weak Galerkin methods are 1) the approximating functions are discontinuous, and 2) the usual derivatives are taken as distributions or approximations of distributions.The method was successfully applied to the second order elliptic equations [8] [9], the Stokes equations [10], Parabolic equations [11], and Maxwell equations [12].A posteriori error is effectively estimated, and proved the convergence of the WG finite element method in this paper.

Preliminaries
In this paper, we study the initial-boundary value problems of the Stokes.
where ( ) , u u u = is fluid velocity, p is pressure, ( ) The solution of the Stokes equations forms an important aspect of both theoretical and computational fluid dynamics.A limited number of solutions of these non-linear partial differential equations mostly involving spatially one-dimensional problems are given in the literature.Solutions of practical interest have been obtained for cases where, with suitable approximations, the equations are reduced to linear partial differential equations.
Let Ω be a bounded domain in R 2 .We introduce function spaces ( ) the unsteady Stokes problem would take the following form: seek ( ) We use

⋅ is
given by and ⋅ is said to be the norm of 2 L .
For w is [ ] The space ( ) s H D and the norm defined in the ( )

Weak Galerkin Finite Element Approximation Scheme
Let K be any polygonal or polyhedral domain with boundary K ∂ .A weak vector-valued function on the region K refers to a vector-valued function ( ) . The first component 0 v can be understood as the value of v in K, and the second component ( ) { } ( ) ( ) Definition 1.For any , the weak gradient of v is defined as a linear functional w v ∇ in the dual space of ( )  , whose action on each ( ) where n is the outward normal direction to K ∂ , ( ) ( ) The Sobolev space ( ) With the help of the inclusion map iν , the Sobolev space ( )  can be viewed as a subspace of ( ) K ν by identifying each ( ) r P K be the set of polynomials on K with degree no more than r.
for all ( ) In what follows, we give the definition of weak divergence, first of all, we require weak function V K the space of weak vector-valued functions on K; Definition 3.For any where n is the outward normal direction to K ∂ , ( ) The Sobolev space ( ) for all ( )

Weak Galerkin Finite Element Scheme
Let h  be a partition of the domain Ω with mesh size h that consists of arbitrary polygons/polyhedra.In this paper, we assume that the partition h  is WG shape regular-defined by a set of conditions as detailed in references.Denote by h ε the set of all edges/flat faces in h  , and let 0 h h ε ε = ∂Ω be the set of all interior edges/faces.For any integer 1 k ≥ , we define a weak Galerkin finite element space for the velocity variable as follows, We would like to emphasize that there is only a single value b v defined on each edge h e ε ∈ .For the pressure variable, we have the following finite element space Denote by 0 h V the subspace of h V consisting of discrete weak functions with vanishing boundary value; The discrete weak gradient V can be computed by using ( 5) and ( 8) on each element T, respectively.More precisely, they are given by , , For simplicity of notation, from now on we shall drop the subscript for the discrete weak gradient and the discrete weak divergence.The usual 2 L inner product can be written locally on each element as follows For each edge/face h e ε ∈ , denote by b Q the L 2 projection from ( ) We are now in a position to describe a weak Galerkin finite element scheme for the Stokes Equations (1).To this end, we first introduce three bilinear forms as follows ( ) ,0 In the following, the proof process of Lemma 1-6 refers to reference [10] [11] [12].
Lemma 1.For any defined in the previous section, let h Q and h Q be two local L 2 projections onto ( ) Lemma 3. The projection operators h Q , h R , and h S satisfy the following commutative properties There exists a positive constant β independent of h such that ( ) Lemma 5. Poincare inequality of Weak gradient operator: If First of all, we study the existence and uniqueness of the solution for (9).The space defined as follows ( ) ( )
Then we need to seek ( ) ( ) ,0 , h u x t be the solution of (10) and which is unique, the linear bounded functional ( ) Then problem ( 9) is equivalent to seek Using LBB condition and Lax-Milgram Lemma, we know that the solution 12) is unique.
Combing ( 11) and ( 12), it is concluded that if initial approximation In what follows, we introduce Stokes projection, which is the important approximation of projection.Lemma 6.First of all, we introduce Stokes projection of ( ) , , ,

Error Equations
In what follows, we list Lemma 7 to prove the error estimation of approximate solution for Semi-discrete scheme.
We know that ( )

and ( )
, Similarly, the pressure p is projected into h W as h S p .Denote by h e and h ε the corres- ponding error given by , be sufficiently smooth and satisfy the following equation t w w S ρ be the L 2 projection of ( ) , w p into the finite element space h h V W × .Then, the following equation holds true , , Proof.Together Lemma 3, Equation ( 5) and integration by parts.we obtain Next, Combing Lemma 3 and Equation ( 8), the fact that , 0 , then using integration by parts, we obtain We can imply that Next, we test (17) by using 0 It follows from the usual integration by parts that ( ) ( ) Where we have used the fact that , 0 . using Equations ( 19) and (20), we , , , Substituting ( 20), ( 22) and ( ) ( ) , , , , In what follows, we give the derivation of the error equation of ( 9).Lemma 8. Let h e and h ε be the error of the weak Galerkin finite element solution arising from ( 9), as defined by ( 16).Then, we The difference of ( 24) and ( 9) yields the following equation, . This completes the derivation of (23).
As to (24), we test Equation (1) by h q W ∈ and use (9) to obtain The difference of (25) and ( 9) yields the following equation

( )
, 0 h b e q = for all h q W ∈ .
In the following, the proof process of Lemma 9 refers to reference [10].
, with the precondition of regular-shape h  , we have the following estimation.

Error Estimates
The following theorem is the main result of this paper.

∫
This completes the proof.Thus, the error estimates of Theorem 1 hold.Optimal-order error estimates are established for the corresponding numerical approximation in an H 1 norm for the velocity, and L 2 norm for both the velocity and the pressure by use of the Stokes projection.

bv
represents v on the boundary of K.Note that b v may not necessarily be re- lated to the trace of 0 v on K ∂ should a trace be well-defined.Denote by ( ) K ν the space of weak functions on K;

∇
on the finite element space h By using Lemma 2, Lemma 5 and Lemma 9, we obtain